Realizing Computably Enumerable Degrees in Separating Classes

TL;DR

This paper explores which collections of computably enumerable (c.e.) Turing degrees can be represented by separating classes, demonstrating realizability for certain degrees and ruling out specific configurations.

Contribution

It shows that for every c.e. degree, the pair with zero and the degree itself can be realized as a separating class, and it rules out the existence of super-maximal pairs with certain properties.

Findings

The collection {c, 0'} can be realized as a separating class.

No super-maximal pairs with infinite separating classes exist.

Certain configurations of c.e. degrees cannot be realized as separating classes.

Abstract

We investigate what collections of c.e.\ Turing degrees can be realised as the collection of elements of a separating $\Pi^0_1$ class of c.e.\ degree. We show that for every c.e.\ degree $\mathbf{c}$, the collection $\{\mathbf{c}, \mathbf{0}'\}$ can be thus realized. We also rule out several attempts at constructing separating classes realizing a unique c.e.\ degree. For example, we show that there is no \emph{super-maximal} pair: disjoint c.e.\ sets $A$ and $B$ whose separating class is infinite, but every separator of c.e.\ degree is a finite variant of either $A$ or $\overline{B}$.